Supplementary Material
Highly luminescent 2D excitons in atomically thin CdSe
nanosheets
1
O. Halder1, A. Pradhani1, P. K. Sahoo2, B. Satapati3 and S. Rath1*
School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Toshali Bhawan,
Satyanagar, Bhubaneswar-751 007, India
2
Department of Physics, National Institute of Science Education and Research,
Sachivalaya marg, Bhubaneswar-751 005
3
Saha Institute of Nuclear Physics, Salt Lake, Kolkata
(a)
Optimization of the synthesis protocol through optical absorption
Fig. S1. Optical absorption spectra of CdSe ASNSs (a) sample prepared at temperature 403 K
for 24hrs (open red triangles), and (b) sample prepared at temperature 393 K for 48hrs (filled
black triangles).
The absorption spectrum of sample –(a), shows multiple absorption peak profiles
revealing the non-uniformity in shapes and sizes which may be due to the instability in the
lamellar geometry. The absorption spectrum of sample – (b) exhibits broad absorption band
followed by a long tailing to the low energy side which may be due to the size distribution.
Note that, an optimum condition has been achieved at 393 K for 24 hrs. The importance of
1
the synthesis route is that we have followed a simple technique avoiding the complex reaction
and co-surfactants.
(b)
Wide angle x-ray scattering (WAXS) studies.
Fig. S2. Wide angle x-ray scattering of the CdSe ASNSs .
It shows the hexagonal structure of the sample with lattice spacing, a = 3.73 Å and c = 6.41
Å in comparison with bulk JCPDS-77-2307 (a = 4.20 nm and c = 7.01 Å).
(c)
Small angle X-ray scattering (SAXS) Analysis
The radial electron density (RED) has been estimated from the SAXS results using the
expression
1

 (r ) 
I (q)  4A p(r )
0
1
2 2

 [ I (q)
1
2
cos qr ]dq
where
the
de-smeared
intensity
0
cos qr
dr , the pair distance distribution function (PDDF) p(r )   (r )  r 2 ,
2
q
2
area of lamellae, A, the normalized 1D autocorrelation function,  (r ) 

1 2
q I (q) cos( qr )dq
Q 0
, r is the coordination along which the electron density distribution, q is the scattering vector

and Q   q 2 I (q)dq . Generalized inversion Fourier transformation (GIFT) technique is used
0
to extract the PDDF profile from the scattering data by fitting with theory1. To quantify the
layer structure, the relative RED has been estimated by spectral fitting of the PDDF profile
using DECON software as per the above theory1.
(d) Optical characterization using the standard set up.
The optical absorption, photoluminescence (PL) and time resolved photoluminescence (TRPL) studies have been carried out using Perkin Elmer UV-VIS spectrophotometer, EdinburgF-900 photoluminescence setup and Horiba Jobin Yvon –FluoroCube time resolved
photoluminescence system coupled with pulse laser (wave length ~ 375 nm and pulse rate ~
200 ps) at room temperature (300 K) respectively.
(e)
Calculation of Exciton binding energy in Nanosheets
The Schrödinger equation for an exciton in a quantum well (QW) can be written as

e2 


 K e  Kh  Ve  z e   Vh  zh     exc  Eexc ,
 re  rh 

(1)
2
 e ,h , the Laplacian  e,h depends on electron, hole coordinates,
where Kˆ e ,h  
2 m e ,h
respectively, Ve,h z e,h  is the confining potential for electron (hole) in the growth z-axis of the
structures.
The above equation has been solved using a trial function
exc  F  R f   U e  ze U h  zh  ,
(2)
 me re  mh rh
 

where R 
is the exciton center of mass,    e   h is the in-plane radiusme  mh


vector of electron and hole and relative motion r   , z  . The trial function (equation-2)
3
contains the exciton center of mass motion, the relative electron-hole motion in the plane of
the QW, electron and hole motion in normal to the plane direction, respectively.
The factorization of the exciton wave function given in equation-2 makes sense when the QW
width is less or comparable to the bulk exciton Bohr diameter of the semiconductor. In this
case, electron and hole are quantized independently of each other. On the other hand, in a
larger QW, one can assume that the exciton is confined as a whole particle and keeps the
internal structure of a 3D hydrogen atom. Here and further we shall consider narrow QWs
where Eq. (2) represents a good approximation. The four terms which compose the exciton
wave-function are normalized as follows:
 dz

e

U e ze    dzh U h zh    2d f     2RdR F R   1 .
2
2
2
0
2
(3)
0

After substitution of the trial function (2) and integration over R , Eq. (1) becomes:


e2
 z z 

K

K

K

V
z

V
z


E

K
 e
h

e e
h h
exc  f   U e  z e U h  z h   0 (4)
2

  2   ze  zh 



Where K exc 
2
Pexc
 2 
 2 
, K ez  
, K hz  
,
2me  mh 
ze 2me ze
zh 2mh zh
1   2  
K   
 ,

   2   

me mh
, Pexc is the excitonic momentum, Pexc  0 for the ground state. Then, Eq. (4)
me  mh
can be transformed into a system of three coupled differential equations each defining one of
the components of our trial function.
The equation for
f    is obtained by multiplication of both parts of Eq. (4) by
U *e  z e U *h  z h  and integration over z e and z h .

e2

Thus,  Kˆ  



2
2
U e z e  U h z h  

QW
 dze dzh  2  z  z 2  f     E B f   ,
e
h

(5)
Where, EBQW is the exciton binding energy.
Solving (5) with a trial function f   
2 1
exp   a , where a is a variational
 a
parameter, one can express the binding energy as:
4
E
QW
B
E
2
2
a     2  e

2a
QW
B
2
e2
a    2 

2a


 dz  dz  2d
e

h

f   U e  z e  U h  z h 
2

0
2
  z e  z h 
2
2
2
.
(6)

4
2
2e 2
 2 
0 d 2 exp   a d   2a 2  a
2
2  2 
2
2 2






2a 2 a  a B 
2a 2 aa B
2  1 4
 1 4
 2    E B3 D  2  
If a  taB , then E BQW 
2 
t
t
2 a B  t
 t
 1 4
Hence, E BQW inE B3 D unit   2  ina B unit 
t
 t
(7)
E BQW a   


(8)
(9)
(10)
References:
1
2
O. Glatter, and O. Kratky, Small angle x-ray Scattering, (Academic Press, London, 1982).
L. M. Ramaniah, S. V. Nair, Phys. Rev. B 47, 7132 (1993).
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